Denef–Lipshitz conjecture on integral Diophantine extensions

Let $L$ be a number field. An extension $L/K$ is integrally diophantine if the ring of integers $\mathcal{O}_K$ is a diophantine subset of $\mathcal{O}_L$, and a subset is diophantine when it is existentially definable by polynomial equations over the ambient ring. The Denef–Lipshitz conjecture asserts that

Denef–Lipshitz conjecture. For every number field $L$, the extension

$$L/\mathbb{Q}$$

is integrally diophantine; equivalently, $\mathbb{Z}=\mathcal{O}_{\mathbb{Q}}$ is a diophantine subset of $\mathcal{O}_L$.

This would imply a negative answer to Hilbert's tenth problem for the rings of integers of all number fields. The conjecture is attributed to Denef and Lipshitz and is unresolved in general.